Since no information was available about the size of source range being copied
Following grey areas of the question is assumed as follows
Since 191 Rows X 68 copy X 3 columns take around 10 minutes only (with you code), the range is about 191 Rows X 15 Columns in size
since it has been claimed that code is working Correctly. The Cells of the range (irrespective of their row or column positions) is being copied in column A only (below one and another). Though it contradicts the statement "automate creation of a data table"
Since the cells of the ranges are being copied and pasted. In test case formulas are copied only.
So the code below will just replicate what your code is doing with some increased efficiency. As I personally don't prefer to keep calculations, event processing and screen updating off (in normal cases) i haven't added that standard lines. However you may use these standard techniques, depending on the working file condition. Make necessary changes regarding Range etc
Code takes only 2-3 seconds to complete with 191 Rows X 15 columns X 68 Copies:
Sub test()
Dim SrcWs As Worksheet, DstWs As Worksheet, SrcArr As Variant
Dim Rng As Range, cell As Range, DstArr() As Variant
Dim X As Long, Y As Long, Z As Long, i As Long, LastRow As Long
Dim Chunk60K As Long
Dim tm As Double
tm = Timer
Set SrcWs = ThisWorkbook.Sheets("Input Data")
Set DstWs = ThisWorkbook.Sheets("TrialSheet")
Set Rng = SrcWs.Range("A1:O191")
SrcArr = Rng.Formula
LastRow = DstWs.Cells(Rows.Count, "A").End(xlUp).Row + 1
Chunk60K = 0
Z = 1
For X = 1 To UBound(SrcArr, 1)
For Y = 1 To UBound(SrcArr, 2)
For i = 1 To 68
ReDim Preserve DstArr(1 To Z)
DstArr(Z) = SrcArr(X, Y)
If Z = 60000 Then ' To Overcome 65K limit of Application.Transpose
DstWs.Range("A" & Chunk60K * 60000 + LastRow).Resize(UBound(DstArr, 1), 1).Formula = Application.Transpose(DstArr)
Chunk60K = Chunk60K + 1
Z = 1
ReDim DstArr(1 To 1)
Debug.Print "Chunk: " & Chunk60K & " Seconds Taken: " & Timer - tm
Else
Z = Z + 1
End If
Next i
Next Y
Next X
If Z > 1 Then DstWs.Range("A" & Chunk60K * 60000 + LastRow).Resize(UBound(DstArr, 1), 1).Formula = Application.Transpose(DstArr)
Debug.Print "Seconds Taken: " & Timer - tm
End Sub